# Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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### Convex planar regions with all area bisectors having equal length

Following A claim on the concurrency of area bisectors of planar convex regions, let me record a couple of simple queries. An area bisector (perimeter bisector) of a planar convex region is a chord ...
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### A rational distance problem with (possibly) multiple solutions

Background: It is not known if any point exists on the XY plane that is at rational distance from all 4 vertices of a unit square lying on the XY plane (https://mathworld.wolfram.com/...
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### Show that a region in a plane defined by a polynomial contains integer points

Let $F \in \mathbb{Z}[x,y]$ be a polynomial of degree $2n$ such that the homogeneous degree $2n$ part of $F$, say $F_{2n}$, is positive semi-definite. How does one show that for some $\delta_n > 0$ ...
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### The closest ellipse to a given triangle

Definition: The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set. Question: Given a general triangle T, to ...
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### On 'axiality' of planar convex regions

Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry) Consider an alternative definition of axiality as follows: For a convex region C, consider a ...
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### Do cut-length-minimizing equidissections exist?

Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
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### Is "Escherian metamorphosis" always possible?

$\DeclareMathOperator\int{int}\DeclareMathOperator\diam{diam}\DeclareMathOperator\area{area}\DeclareMathOperator\cl{cl}\DeclareMathOperator\ran{ran}\DeclareMathOperator\dom{dom}$This is a tweaked ...
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### Minimize total area bounded by $N$ lines in general position

Suppose we have $N$ lines in general position (any two lines, but no three lines, meet at a point) ($N\geq 3$). Let the smallest bounded region have area $1$. Determine the minimum (or possibly ...
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### Ellipse of least perimeter that contains a given triangle

This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue. Question: Given a general triangle T, How does one find and characterize the ellipse of least ...
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